Differential-algebraic Equations Research Articles

New approach in solving derivative causality problems in the bond-graph method

From the bond-graph model of a dynamic system, the state equations can be obtained. When the system contains only energy storing elements in integral causality, the system of state equations is immediately and directly determined. When the model contains at least one energy storing element in derivative causality, the resulted system of differential-algebraic equations arises some difficulties in finding the final form of the system of state equations. The work presents a new method of deducing the state equations in case of bond-graph models with one, or several energy storing elements in derivative causality. This method is based on the kinetic energy of the system and offers the possibility to avoid a difficult mathematical calculus for the transition from a system of differential algebraic equations to the system of state equations.

Dynamic modeling of a topside process plant with modified black-oil approach

Modeling and simulation of oil and gas facilities has been done by several authors in closed-source commercial applications, with low flexibility for model adaptation. A few authors used free modeling environments, which results in a differential-algebraic equations (DAE) system and lets the model be adaptable for plantwide simulation. Still, it has limitations in obtaining the model's initial conditions and to perform dynamic simulations when the compositional approach is used. The Modified Black-Oil (MBO) approach, which models the thermodynamic equilibria in a simpler way, has been used by several authors for reservoir modeling but not for topside facilities. This work proposes the use of the MBO approach for topside dynamic modeling. This was done in EMSO©, obtaining a large index-1 DAE system and successful obtention of consistent initial conditions. Disturbances on specific variables were applied and shows the potential use of the MBO approach for new dynamic studies.

On Existence and Uniqueness of Formal Power Series Solutions of Algebraic Ordinary Differential Equations

Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when a truncated power series can be extended to a formal power series solution. If a certain regularity condition on the given AODE or on the initial values is fulfilled, we compute all of the solutions. Moreover, when the existence is confirmed, we present the algebraic structure of the set of all formal power series solutions.

Singular perturbation results for linear partial differential–algebraic equations of hyperbolic type

We consider constrained partial differential equations of hyperbolic type with a small parameter ε>0, which turn parabolic in the limit case, i.e., for ε=0. The well-posedness of the resulting systems is discussed and the corresponding solutions are compared in terms of the parameter ε. For the analysis, we consider the system equations as partial differential–algebraic equation based on the variational formulation of the problem. For a particular choice of the initial data, we reach first- and second-order estimates. For general initial data, lower-order estimates are proven and their optimality is shown numerically.

Multi-resolution lattice Green's function method for incompressible flows

We propose a multi-resolution strategy that is compatible with the lattice Green's function (LGF) technique for solving viscous, incompressible flows on unbounded domains. The LGF method exploits the regularity of a finite-volume scheme on a formally unbounded Cartesian mesh to yield robust and computationally efficient solutions. The original method is spatially adaptive, but challenging to integrate with embedded mesh refinement as the underlying LGF is only defined for a fixed resolution. We present an ansatz for adaptive mesh refinement, where the solutions to the pressure Poisson equation are approximated using the LGF technique on a composite mesh constructed from a series of infinite lattices of differing resolution. To solve the incompressible Navier-Stokes equations, this is further combined with an integrating factor for the viscous terms and an appropriate Runge-Kutta scheme for the resulting differential-algebraic equations. The parallelized algorithm is verified through with numerical simulations of vortex rings, and the collision of vortex rings at high Reynolds number is simulated to demonstrate the reduction in computational cells achievable with both spatial and refinement adaptivity.

Multiscale modeling and nonlinear model predictive control for flue gas desulfurization

The primary source of sulfur dioxide (SO2) emissions is flue gas from fossil fuels-based power plants. SO2 emissions are known to not only cause health issues, but also have an adverse effect on the environment in various ways. Several Flue Gas Desulfurization (FGD) technologies have been incorporated in power plants. The most popular technology is Wet FGD, where a limestone slurry is used to absorb SO2 from the flue gas. A detailed droplet scale model describing the instantaneous and finite rate chemistry is developed. The ill-posed Differential-Algebraic Equation (DAE) droplet model is reformulated to a well-posed index-1 DAE through index reduction. The droplet model is integrated with the bulk phase by incorporating gas-liquid mass transfer, and an oxidation reactor model to simulate the dynamic operation of the counter-current spray scrubber. As a result, the model has a well-conditioned Jacobian and overcomes the modeling challenges of previous works and enables numerical solution without requiring carefully selected initialization or specialized solution procedures. The model is successfully validated using power plant measurements, and nonlinear model predictive control (NMPC) studies are demonstrated to optimize recycle stream flowrates to minimize pumping costs.

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

Computer-assisted proofs for some nonlinear diffusion problems

In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form Δu+f(u,∇u)=0. By studying a well chosen fixed point problem defined around the numerical solution, these techniques make it possible to prove the existence of a solution in an explicit (and usually small) neighborhood of the numerical solution. In this work, we develop a similar approach for a broader class of systems, including nonlinear diffusion terms of the form ΔΦ(u). In particular, this enables us to obtain new results about steady states of a cross-diffusion system from population dynamics: the (non-triangular) SKT model. We also revisit the idea of automatic differentiation in the context of computer-assisted proof, and propose an alternative approach based on differential–algebraic equations.

Optimal controller design for non-affine nonlinear power systems with static var compensators for hybrid UAVs

A generalized non-affine nonlinear power system model is presented for a single machine bus power system with a Static Var Compensator (SVC) or State Var System (SVS) for hybrid Unmanned Aerial Vehicles (UAVs). The model is constructed by differential algebraic equations on the MATLAB-Simulink platform with the programming technique of its S-Function. Combining the inverse system method and the Linear Quadratic Regulation (LQR), an optimized SVC controller is designed. The simulations under three fault conditions show that the proposed controller can effectively improve the power system transient performance.

An analysis of structural and practical identifiability applied to a transient generator model

This article addresses issues of the structural and practical identifiability of a transient model of a synchronous generator. A structural identifiability analysis was conducted for nonlinear models of differential-algebraic equations (DAEs) and then applied to the DAE model of the synchronous generator, whose parameters were estimated by a practical identifiability analysis employing a trajectory sensitivity method. An optimal time window of measurement data was selected for improving the accuracy of the estimated parameters, which were validated from different sets of measurements collected in two different experiments.

Analytical model of the in-plane torsion test

In research and industry, the in-plane torsion test is applied to investigate the material behaviour at large plastic strains: a sheet is clamped in two concentric circles, the boundaries are twisted against each other applying a torque, and simple shear of the material arises. This deformation is analysed within the scope of finite elasto-plasticity. An additive decomposition of the Almansi strain tensor is derived, valid as an approximation for arbitrary large plastic strains and sufficiently small elastic strains and rotations. Constitutive assumptions are the von Mises yield criterion, an associative flow rule, isotropic hardening, and a physically linear elasticity relation. The incremental formulation of the elasticity relation applies covariant Oldroyd derivatives of the stress and the strain tensors. The assumptions combined with equilibrium conditions lead to evolution equations for the distribution of stresses and accumulated plastic strain. The nonzero circumferential stress must be determined from the equilibrium condition because no deformation is present in tangential direction. As a result, a differential-algebraic-equation (DAE) system is derived, consisting of three ordinary differential equations combined with one algebraic side condition. As an example material, properties of a dual phase steel DP600 are analysed numerically at an accumulated plastic strain of 3.0. Radial normal stresses of 3.1% and tangential normal stresses of 1.0% of the shear stresses are determined. The influence of the additional normal stresses on the determination of the flow curve is 0.024%, which is negligibly small in comparison with other experimental influences and measurement accuracies affecting the experimental flow curve determination.

An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problems

Navier-Stokes equations are well known in modeling of an incompressible Newtonian fluid, such as air or water. This system of equations is very complex due to the non-linearity term that characterizes it. After the linearization and the discretization parts, we get a descriptor system of index-2 described by a set of differential algebraic equations (DAEs). The two main parts we develop through this paper are focused firstly on constructing an efficient algorithm based on a projection technique onto an extended block Krylov subspace, that appropriately allows us to construct a reduced system of the original DAE system. Secondly, we solve a Linear Quadratic Regulator (LQR) problem based on a Riccati feedback approach. This approach uses numerical solutions of large-scale algebraic Riccati equations. To this end, we use the extended Krylov subspace method that allows us to project the initial large matrix problem onto a low order one that is solved by some direct methods. These numerical solutions are used to obtain a feedback matrix that will be used to stabilize the original system. We conclude by providing some numerical results to confirm the performance of our proposed method compared to other known methods.

FORMATION OF A DESIGN MODEL AND ANALYSIS OF THE STRESS-STRAIN STATE OF STRUCTURAL ELEMENTS UNDER ALTERNATING LOADING

The article gives geometric and physical relationships for shell structure elements. To describe the elastoplastic behavior of the shell element, the theory of variable plasticity is used, which allows one to take into account hardening-softening, cyclic anisotropy, and material damage. Applying the linearization method, after a series of transformations, a canonical system of differential-algebraic equations is obtained that describes the elastoplastic behavior of a composite shell structure. Boundary-value problems are solved using the orthogonal sweep method.

Renormalized Oscillation Theory for Linear Hamiltonian Systems on [0, 1] Via the Maslov Index

Working with a general class of regular linear Hamiltonian systems, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of \({\mathbb {C}}^{2n}\). We verify that our applicability class includes Dirac and Sturm–Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly.

Geometric analysis of nonlinear differential-algebraic equations via nonlinear control theory

For nonlinear differential-algebraic equations (DAEs), we define two kinds of equivalences, namely, the external and internal equivalence. Roughly speaking, the word “external” means that we consider a DAE (locally) everywhere and “internal” means that we consider the DAE on its (locally) maximal invariant submanifold (i.e., where its solutions exist) only. First, we revise the geometric reduction method in DAEs solution theory and formulate an implementable algorithm to realize that method. Then a procedure called explicitation with driving variables is proposed to connect nonlinear DAEs with nonlinear control systems and we show that the driving variables of an explicitation system can be reduced under some involutivity conditions. Finally, due to the explicitation, we use some notions from nonlinear control theory to derive two nonlinear generalizations of the Weierstrass form.

An α-Model Parametrization Algorithm for Optimization with Differential-Algebraic Equations

An optimization task with nonlinear differential-algebraic equations (DAEs) was approached. In special cases in heat and mass transfer engineering, a classical direct shooting approach cannot provide a solution of the DAE system, even in a relatively small range. Moreover, available computational procedures for numerical optimization, as well as differential- algebraic systems solvers are characterized by their limitations, such as the problem scale, for which the algorithms can work efficiently, and requirements for appropriate initial conditions. Therefore, an αDAE model optimization algorithm based on an α-model parametrization approach was designed and implemented. The main steps of the proposed methodology are: (1) task discretization by a multiple-shooting approach, (2) the design of an α-parametrized system of the differential-algebraic model, and (3) the numerical optimization of the α-parametrized system. The computations can be performed by a chosen iterative optimization algorithm, which can cooperate with an outer numerical procedure for solving DAE systems. The implemented algorithm was applied to solve a counter-flow exchanger design task, which was modeled by the highly nonlinear differential-algebraic equations. Finally, the new approach enabled the numerical simulations for the higher values of parameters denoting the rate of changes in the state variables of the system. The new approach can carry out accurate simulation tests for systems operating in a wide range of configurations and created from new materials.

An assessment of implicit-explicit time integrators for the pseudo-spectral approximation of Boussinesq thermal convection in an annulus

We analyze the behavior of an ensemble of time integrators applied to the semi-discrete problem resulting from the spectral discretization of the equations describing Boussinesq thermal convection in a cylindrical annulus. The equations are cast in their vorticity-streamfunction formulation that yields a differential algebraic equation (DAE). The ensemble comprises 28 members: 4 implicit-explicit multistep schemes, 22 implicit-explicit Runge-Kutta (IMEX-RK) schemes, and 2 fully explicit schemes used for reference. The schemes whose theoretical order varies from 2 to 5 are assessed for 11 different physical setups that cover laminar and turbulent regimes. Multistep and order 2 IMEX-RK methods exhibit their expected order of convergence under all circumstances. IMEX-RK methods of higher-order show occasional order reduction that impacts both algebraic and differential field variables. We ascribe the order reduction to the stiffness of the problem at hand and, to a larger extent, the presence of the DAE. Using the popular Crank-Nicolson Adams-Bashforth of order 2 (CNAB2) integrator as reference, performance is defined by the ratio of maximum admissible time step to the cost of performing one iteration; the maximum admissible time step is determined by inspection of the time series of viscous dissipation within the system, which guarantees a physically acceptable solution. Relative performance is bounded between 0.5 and 1.5 across all studied configurations. Considering accuracy jointly with performance, we find that 6 schemes consistently outperform CNAB2, meaning that in addition to allowing for a more efficient calculation, the accuracy that they achieve at their operational, dissipation-based limit of stability yields a lower error. In our most turbulent setup, where the behavior of the methods is almost entirely dictated by their explicit component, 13 IMEX-RK integrators outperform CNAB2 in terms of accuracy and efficiency.

State for Linear Time-Varying Systems, with Applications to Dissipative Systems

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 7 July 2020Accepted: 08 October 2021Published online: 10 January 2022Keywordsstate variable, differential-algebraic equations with time-varying coefficients, dissipativity, storage function, bilinear and quadratic differential formsAMS Subject Headings93A10, 93A30, 93B25, 93C05Publication DataISSN (print): 0363-0129ISSN (online): 1095-7138Publisher: Society for Industrial and Applied MathematicsCODEN: sjcodc

Modeling and control of anterior–posterior and medial–lateral sways in standing posture

To study essential anterior–posterior and medial–lateral sways of the stance caused by rotational movements about the ankle and hip joints, a mathematical model is developed for the 3D postural kinematics and dynamics. The model is in the form of nonlinear differential–algebraic equations corresponding to a biomechanical system with holonomic constraints. A nonlinear feedback control law is further derived for stabilizing the upright stance, whilst eliminating internal torques induced by the constraints on postural movements. Numerical simulations of the model parametrized with experimental data of human body segments illustrate the performance of postural balancing with the proposed control.

A cell-based PBM for continuous open-circuit dry milling: Impact of axial mixing, nonlinear breakage, and screen size

This theoretical study examined the impact of the degree of mixing, nonlinear particle breakage, and screen opening size on the particle size distribution (PSD) and mass hold-up in continuous dry mills with internal classification. A cell-based population balance model (PBM) incorporating a non-ideal screen model was formulated, wherein the back-mixing ratio and number of cells modulated the extent of axial mixing. The set of differential–algebraic equations (DAEs) was solved for the spatio-temporal evolution of the PSD in the mill and the product stream. The simulation results suggest that a smaller screen opening delayed the attainment of the steady state, increased the hold-up, and yielded a finer product PSD. The cushioning action of fines resulted in a coarser product PSD; however, a screen with a smaller opening mitigated this effect. The cell-based PBM predicted various features of experimental milling observations while providing insights into the mixing–nonlinear breakage–classification interplay.